Operator algebra seminar

Speaker: Marcus de Chiffre (TU Dresden)

Title: Stability results for $\varepsilon$-representations

Abstract: Let $\varepsilon>0$, let $G$ be a group and let $\mathcal M$ be a von Neumann algebra with unitary group $\mathcal U(\mathcal M)$. An $\varepsilon$-representation of $G$ (with respect to $\Vert\cdot \Vert$) is a map $\varphi\colon G\to\mathcal U(\mathcal M)$ such that $\Vert\varphi(gh)-\varphi(g)\varphi(h)\Vert<\varepsilon$ for all $g,h\in G$. Here, $\Vert\cdot\Vert$ is some norm on $\mathcal M$, e.g.\ the operator norm or, if $\mathcal M$ has a tracial state, the 2-norm.

The topic of this talk is the following question: Given an $\varepsilon$-representation as above, is there an honest representation $\pi\colon G\to\mathcal U(\mathcal M)$ such that the quantity $\Vert\varphi(g)-\pi(g)\Vert$ is small (depending on $\varepsilon$) for all $g\in G$?
We will answer this question in various constellations with focus on the case where $\mathcal M$ is a finite factor equipped with the 2-norm.